A057000 -id:A057000 - cp's OEIS frontend

A229552 Numbers k such that phi(k) = phi(k+2) - phi(k+1).

1, 3, 5, 9, 15, 21, 35, 39, 45, 99, 135, 231, 255, 855, 1035, 1295, 1539, 1599, 2015, 4335, 6525, 9177, 14399, 16095, 30495, 55385, 61131, 62799, 65535, 72579, 77615, 110175, 152649, 179295, 244965, 299649, 603459, 619695, 686735, 1876725, 2841915, 3058209

Offset: 1

Vincenzo Librandi, Sep 27 2013

nonn

Donovan Johnson and Giovanni Resta, Table of n, a(n) for n = 1..324 (terms < 10^13, first 100 terms from Donovan Johnson)

Cf. A092404, A197112, A057000, A000010, A065557.

Select[Range[10^6], EulerPhi[#] == EulerPhi[# + 2] - EulerPhi[# + 1] &]
Position[Partition[EulerPhi[Range[31*10^5]],3,1],?(#[[3]]-#[[2]] == #[[1]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Nov 29 2017 *)

a(n) = A065557(n) - 2. - Amiram Eldar, Dec 09 2022

A250222 a(n) = phi(2n+1) - phi(2n), where phi is A000010.

Original entry on oeis.org

1, 2, 4, 2, 6, 8, 2, 8, 12, 4, 12, 12, 6, 16, 22, 4, 8, 24, 6, 24, 30, 4, 24, 26, 12, 28, 22, 12, 30, 44, 6, 16, 46, 12, 46, 48, 4, 24, 54, 22, 42, 40, 14, 48, 48, 16, 26, 64, 18, 60, 70, 0, 54, 72, 32, 64, 52, 16, 38, 78, 20, 40, 90, 20, 82, 68, 6, 72, 94, 44, 50, 64

Offset: 1

Eric Chen, Dec 24 2014

a(157) = -12 is the first negative number in this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A037225, A057000, A062570, A185199.

a[n_] := EulerPhi[2*n+1] - EulerPhi[2*n]; Array[a, 100] (* Amiram Eldar, Nov 09 2024 *)
#[[2]]-#[[1]]&/@Partition[EulerPhi[Range[2,150]],2] (* Harvey P. Dale, Aug 02 2025 *)

a(n) = eulerphi(2*n+1) - eulerphi(2*n); \\ Amiram Eldar, Nov 09 2024

From Amiram Eldar, Nov 09 2024: (Start)
a(n) = A057000(2*n).
a(n) = A037225(n) - A062570(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 4/Pi^2 = 0.405284... (A185199). (End)

A331573 The bottom entry in the forward difference table of the Euler totient function phi for 1..n.

Original entry on oeis.org

1, 0, 1, -2, 5, -14, 39, -102, 247, -558, 1197, -2494, 5167, -10850, 23311, -51132, 113333, -250694, 547871, -1175998, 2475153, -5117486, 10439895, -21142030, 42777735, -86960284, 178221401, -368541508, 767762191, -1606535062, 3365499467, -7038925364, 14671422797, -30450115592

Offset: 1

Robert G. Wilson v, Jan 20 2020

sign

a(2n) is a nonpositive even number while a(2n-1) is an odd positive number.

Abs(a(n)) < abs(a(n+1)) for 1 < n < 8000.

			a(8) = -102 because:
1     1     2     2     4     2     6     4  (first 8 terms of A000010)
   0     1     0     2    -2     4    -2     (first 7 terms of A057000)
      1    -1     2    -4     6     6
        -2     3    -6    10   -12
            5    -9    16   -22
             -14    25   -38
                 39   -63
                  -102
The first principal right descending diagonal is this sequence.

Cf. A187202, A000010, A057000.

Cf. A002088.

f[n_] := Differences[ Array[ EulerPhi, n], n -1][[1]]; Array[f, 34] (* or *)
nmx = 34; Join[ {1}, Differences[ Array[ EulerPhi, nmx], #][[1]] & /@ Range[nmx - 1]]

a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n-1,k-1)*phi(k). - Ridouane Oudra, Aug 21 2021

a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n,k)*A002088(k). - Ridouane Oudra, Oct 02 2022

A362913 Array of numbers read by upward antidiagonals: leading row lists phi(i), i >= 1 (cf. A000010); the following rows give absolute values of differences of previous row.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 0, 1, 0, 2, 1, 1, 2, 2, 4, 0, 1, 2, 0, 2, 2, 1, 1, 0, 2, 2, 4, 6, 0, 1, 2, 2, 0, 2, 2, 4, 1, 1, 2, 0, 2, 2, 0, 2, 6, 0, 1, 2, 0, 0, 2, 0, 0, 2, 4, 1, 1, 0, 2, 2, 2, 4, 4, 4, 6, 10, 0, 1, 2, 2, 0, 2, 4, 0, 4, 0, 6, 4, 1, 1, 0, 2, 0, 0, 2, 2, 2, 2, 2, 8, 12, 0, 1, 2, 2

Offset: 1

N. J. A. Sloane, May 09 2023

The leading entries in the rows (the Gilbreath transform of A000010, cf. A362451) appear to form the period-2 sequence 1,0,1,0,1,0,... Is there a simple proof? This would follow if there was a proof that the Gilbreath transform of |A057000| is the all-1's sequence.

			The array begins:
  1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, ...
  0, 1, 0, 2, 2, 4, 2, 2, 2, 6, 6, 8, 6, 2, 0, ...
  1, 1, 2, 0, 2, 2, 0, 0, 4, 0, 2, 2, 4, 2, ...
  0, 1, 2, 2, 0, 2, 0, 4, 4, 2, 0, 2, 2, ...
  1, 1, 0, 2, 2, 2, 4, 0, 2, 2, 2, 0, ...
  0, 1, 2, 0, 0, 2, 4, 2, 0, 0, 2, ...
  1, 1, 2, 0, 2, 2, 2, 2, 0, 2, ...
  0, 1, 2, 2, 0, 0, 0, 2, 2, ...
  1, 1, 0, 2, 0, 0, 2, 0, ...
  ...
The first few antidiagonals are:
  1
  0, 1
  1, 1, 2
  0, 1, 0, 2
  1, 1, 2, 2, 4
  0, 1, 2, 0, 2, 2
  1, 1, 0, 2, 2, 4, 6
  0, 1, 2, 2, 0, 2, 2, 4
  1, 1, 2, 0, 2, 2, 0, 2, 6
  0, 1, 2, 0, 0, 2, 0, 0, 2, 4
  ...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened)
N. J. A. Sloane, Maple code for Gilbreath transform and related arrays
N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)

Cf. A000010 (top row of array), A057000 (signed version of second row), A362451,

A362913[dmax_]:=With[{d=Reverse[NestList[Abs[Differences[#]]&,EulerPhi[Range[dmax]],dmax-1]]},Array[Diagonal[d,#]&,dmax,1-dmax]];A362913[20] (* Generates 20 antidiagonals *) (* Paolo Xausa, May 10 2023 *)

A295865 Numbers k, the smallest of at least 4 consecutive numbers x, for which phi(x) <= phi(x+1).

Original entry on oeis.org

1, 2, 14, 104, 164, 254, 494, 584, 1484, 2204, 2534, 2834, 3002, 3674, 3926, 4454, 4484, 4784, 4844, 5186, 5264, 5312, 5984, 6104, 7994, 8294, 8414, 8774, 8834, 9074, 9164, 9944, 10004, 10604, 10724, 11024, 11684, 11894, 12254, 13034, 13064, 13166, 13364, 13454, 13754, 14234, 15344, 15554, 16184, 16214

Offset: 1

Torlach Rush, Feb 13 2018

nonn

There are 3988536 terms below 2*10^9.
Up to a(3988356):
- a(1) is the only odd term.
- a(1) is the only term with 5 consecutive numbers where phi(k) <= phi(k+1).
- the only powers of 2 are a(1)=1 and a(2) = 2.
- of the residues of a(n) mod 10, 4 accounts for greater than 91%.
- if a(n) is divisible by 4, then phi(a(n)) is divisible by 4.
Numbers k such that A057000(k) >= 0 for 3 consecutive terms. - Michel Marcus, Mar 21 2018

			14 is a term because phi(14) <= phi(15) <= phi(16) <= phi(17).
15 is not a term because phi(15) <= phi(16) <= phi(17) > phi(18).

Cf. A000010, A057000.

Phi:= map(numtheory:-phi, [$1..20001]):
DPhi:= Phi[2..-1]-Phi[1..-2]:
C:= select(t -> DPhi[t]>=0, [$1..20000]):
C[select(t -> C[t+2]=C[t]+2, [$1..nops(C)-3])]; # Robert Israel, Mar 26 2018

Drop[#, -2] & /@ Select[SplitBy[#, Last@ # >= 0 &], Length@ # > 2 && #[[1, -1]] >= 0 &][[All, All, 1]] &@ MapIndexed[{First@ #2, #1} &, Differences@ Array[EulerPhi, 2^14]] // Flatten (* Michael De Vlieger, Mar 26 2018 *)

isok(n) = {my(v = vector(4, k, eulerphi(n+k-1))); (v[1] <= v[2]) && (v[2] <= v[3]) && (v[3] <= v[4]);} \\ Michel Marcus, Mar 21 2018

cp's OEIS Frontend

A229552 Numbers k such that phi(k) = phi(k+2) - phi(k+1).

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A250222 a(n) = phi(2n+1) - phi(2n), where phi is A000010.

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A331573 The bottom entry in the forward difference table of the Euler totient function phi for 1..n.

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A362913 Array of numbers read by upward antidiagonals: leading row lists phi(i), i >= 1 (cf. A000010); the following rows give absolute values of differences of previous row.

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A295865 Numbers k, the smallest of at least 4 consecutive numbers x, for which phi(x) <= phi(x+1).

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